Summary: We explore the separability of point sets in the plane by a restricted-orientation convex hull, which is an orientation-dependent, possibly disconnected, and non-convex enclosing shape that generalizes the convex hull. Let \(R\) and \(B\) be two disjoint sets of red and blue points in the plane, and \(\mathcal{O}\) be a set of \(k\geq 2\) lines passing through the origin. We study the problem of computing the set of orientations of the lines of \(\mathcal{O}\) for which the \(\mathcal{O}\)-convex hull of \(R\) contains no points of \(B\). For \(k=2\) orthogonal lines we have the rectilinear convex hull. In optimal \(O(n\log n)\) time and \(O(n)\) space, \(n = \vert R \vert + \vert B \vert\), we compute the set of rotation angles such that, after simultaneously rotating the lines of \(\mathcal{O}\) around the origin in the same direction, the rectilinear convex hull of \(R\) contains no points of \(B\). We generalize this result to the case where \(\mathcal{O}\) is formed by \(k \geq 2\) lines with arbitrary orientations. In the counter-clockwise circular order of the lines of \(\mathcal{O}\), let \(\alpha_i\) be the angle required to clockwise rotate the \(i\)th line so it coincides with its successor. We solve the problem in this case in \(O(1/\Theta \cdot N \log N)\) time and \(O(1/\Theta \cdot N)\) space, where \(\Theta = \min \{ \alpha_1,\ldots, \alpha_k\}\) and \(N=\max \{ k,\vert R \vert + \vert B \vert \}\). We finally consider the case in which \(\mathcal{O}\) is formed by \(k=2\) lines, one of the lines is fixed, and the second line rotates by an angle that goes from 0 to \(\pi\). We show that this last case can also be solved in optimal \(O(n\log n)\) time and \(O(n)\) space, where \(n = \vert R \vert + \vert B \vert\).